Undecidable problem

About: Undecidable problem is a research topic. Over the lifetime, 3135 publications have been published within this topic receiving 71238 citations.

Rapidly Growing Ramsey Functions

Abstract: Godel’s paper on formally undecidable propositions in first order Peano arithmetic (Godel 1931) showed that any recursive axiomatic system containing Peano arithmetic still admits propositions which are not decidable. Godel’s original example of such a proposition was not that illuminating. It was merely a kind of formalization of the well known antinomy of the liar. This raised the problem to look for intuitively meaningful propositions which are independent of Peano arithmetic.

On Some Decision Problems in Programming

Abstract: One of the central problems in programming is the correctness problem, i.e., the question of whether a program computes a given function. We choose a rather general formal semantical framework, effectively given topological T0-spaces, and study the problem to decide whether an element of the space is equal to a fixed element. Moreover, we consider the problems of deciding for two elements, whether they are equal and whether one approximates the other in the specialization order. These are one-one equivalent for a large class of spaces, including effectively given Scott domains. All these problems are undecidable. In most cases they are complete on some level of the arithmetical and/or the Boolean hierarchy. The complexity respectively depends on whether the fixed element is not finite and whether the space contains a nonfinite element. The problem of deciding whether an element is not finite is potentially ?02-complete and for domain-like spaces the membership problem of any nonempty set of nonfinite elements that intersects the effective closure of its complement is ?02-hard. If the given element is finite or the space contains only finite elements, the complexity also depends on the location of the given element in the specialization order and/or the boundedness of the set of lengths of all decreasing chains of basic open sets.

Two-Variable Logic with Two Order Relations

Abstract: It is shown that the finite satisfiability problem for two-variable logic over structures with one total preorder relation, its induced successor relation, one linear order relation and some further unary relations is EXPSPACE-complete. Actually, EXPSPACE-completeness already holds for structures that do not include the induced successor relation. As a special case, the EXPSPACE upper bound applies to two-variable logic over structures with two linear orders. A further consequence is that satisfiability of two-variable logic over data words with a linear order on positions and a linear order and successor relation on the data is decidable in EXPSPACE. As a complementing result, it is shown that over structures with two total preorder relations as well as over structures with one total preorder and two linear order relations, the finite satisfiability problem for two-variable logic is undecidable.